Impedance mechanical electrical

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Original poster: "Jared Dwarshuis" <jdwarshuis@xxxxxxxxx>


Original poster: Jim <<mailto:branley1@xxxxxxxxxxx>branley1@xxxxxxxxxxx >

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Tesla list wrote:
>Original poster: "Jared Dwarshuis" 
<<mailto:jdwarshuis@xxxxxxxxx> jdwarshuis@xxxxxxxxx>
>
>
>Impedance, Mechanical, Electrical
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>
>I wrote this to help the novice coiler
>understand a rather difficult topic. There is
>nothing familiar or intuitive about the
>mathematical approach to understanding
>impedance. Brains are not "wired from the
>factory" for grasping second order differential
>equations and imaginary numbers. Such stuff
>takes practice, and faith! Fortunately there are tangible,
>
>" hands on" physical objects that can be related
>to. This will help us to resolve the mystery.
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>So why should I care about this topic, where is the impetus ?
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>Always a valid question!
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>Hmmm, well??.
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>A Tesla coil should have a power factor correction for the inductive "load"
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>A Tesla coil might need radio frequency suppression, a "choke"
>
>Optimal transfer of energy to the load will
>require careful impedance matching.
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>
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>You need to be armed with some theoretical
>weaponry, or none of this will ever make any sense.
>
>????????.???????????????????????????????????????????????????????????????
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>When we have a mass tied to the end of a spring
>we have an object that behaves mathematically
>very much like a series inductor and capacitor.
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>Capacitors are like springs. Theoretically you
>could store charge forever just like you could
>keep a spring under tension forever. Both cases concern potential energy.
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>Inductors are very much like a mass. Lenz law
>and Newton second law of motion are essentially
>equivalent. Both cases concern forms of kinetic energy. (energy of motion!)
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>Since the lumped model of LCR and dampened
>mass/spring share the exact same energy
>equations, they also share the same equations
>for impedance. The amplitude equations which
>describe how much mass and spring bob up and
>down along the X axis are very old, I don't know
>who first wrote them. Here I will demonstrate
>that the familiar impedance formula can actually
>be derived from mass spring amplitude equations.
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>M dv/dt + CV +KX = F e(iwt) (where C is Kg/s)
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>L di/dt + RI + Q/C = V e(iwt) (where this C is capacitance)
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>Let:
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>X(t) = A e i(wt-phi)
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>Q(t) = Ae i(wt-phi)
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>Then:
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>M d2/dt2 A e i(wt-phi) + C d/dt A e i(wt-phi) + K A e i(wt-phi) = F e iwt
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>Where: M = Kg C = Kg/s K = Kg/ss F = Newton
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>L d2/dt2 A e i(wt-phi) + R d/dt A e i(wt-phi) + 1/C A e i(wt-phi) = V e iwt
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>Where: L = Kg (m/coul)sqrd R = Kg/s (m/coul)sqrd
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>1/C = Kg/ss (m/coul)sqrd V = N (m/coul)
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>Equating the real and imaginary parts of the
>previous equations, squaring both sides, using
>the trigonometric identity: (sine phi) sqrd + (cos phi)sqrd = 1
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>We get:
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>X = F / ((K- M(w)sqrd)sqrd + Csqrd(w)sqrd))^1/2
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>Q = V / ((1/C- L(w)sqrd)sqrd + Rsqrd(w)sqrd))^1/2
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>(w) can be pulled from the denominator.
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>Then:
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>X(w) = Velocity = F/ [ ( K/(w) ­ M(w) )sqrd + Csqrd ]^1/2
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>Q(w) = Current = V/ [ ((w)L ­ 1/(w)C )sqrd + Rsqrd ]^1/2
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>We recognize impedance in the denominator for both systems
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>On the surface it may seem a bit odd that X
>corresponds to Q, but in reality both are just
>describing displacements from an equilibrium
>value. In the case of a dampened mass/spring,
>the equilibrium is simply where the device comes to rest.
>
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>So far a lot of math. But what does it all mean??
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>Lets look at the easiest part first, namely
>resistance. (named C in the mechanical world.)
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>C in the mechanical world has units of kg/s.
>When we push a box across a uniform floor at a
>constant velocity we can describe this with a
>simple force equation. Force = velocity x C The
>units are: Kg m/ ss = m/s x kg/s (C describes how slippery the floor is)
>
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>R in the electrical world is analogous. Voltage
>takes the roll of force. Current becomes
>velocity (moving charge) and R takes the roll of C.
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>*C and R are both linear in these models, not
>always true in either the electric or mechanical worlds.*
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>?????????????????????????............................................................................................................................................................................................ 

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>Now an informal look at reactance!
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>Let us imagine a smooth block attached to a
>spring which is at the other end anchored to a
>wall such that our smooth block can slide on a slightly rough floor.
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>We can always wiggle and jiggle the contraption
>to find exactly when to push, so as to build up pushes for maximum jiggle.
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>Why is that?
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>Well suppose the block is really massive but
>attached to a bitty joke of a spring.
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>Question?
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>Since it is attached to a bitty spring is it ok to give it a good kick?
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>Answer!
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>No! You will break your foot, nearly all of the
>force is going into moving the large mass
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>Ok! Now push the contraption slowly. Did you
>will find that nearly all the force went into the collapsing the spring?
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>This is the essence of reactance!
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>When we found our maximum jiggle, (likely
>without realizing it) we had found the perfect
>compromise point between pushing slowly and
>giving it a good kick. And of course since we
>are talking about different amounts of time
>between pushes, we are actually talking about
>frequency. (And so it is!, reactance is frequency dependant.)
>
>
>

>We have not exhausted the topic, there is also
>the matter of phase angle. Also there are
>several other forms of impedance that we have
>not even talked about. But enough for now.
>
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>End.
>
>Jared Dwarshuis w/07
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>

Jiggle vs Kick, both methods use the same amount
of force I presume. One applies the force
gradually, the other applies the force rapidly. The jiggle conserves energy?
Jim



Hi Jim:



Force for moving the mass comes from accelerating 
the mass (f =ma), acceleration has time components in the denominator (m/ss).



Force from the spring comes from compressing the 
spring KX ( distance times the spring constant) 
the constant K has time components  in the denominator  (kg/ss)



Force from friction comes from velocity x C. Each 
has time components in the denominator. (m/s and kg/s)



Friction cause energy to be lost from the system, 
all the other forms of energy are conserved. The 
units for C, namely kg/s,   suggest a leak of 
mass. Like a sandbag loosing sand as it is tossed 
across a room. By the time the sandbag gets 
across the room it could be nearly empty and have 
very little effective force left.



Our mass and spring are tied together in series. 
So we must consider the force from compressing 
the spring. The force from accelerating the mass. 
And the force from friction separately.



So in the case of no friction with a large mass 
and a tiny spring, we can ignore the spring. In 
the case of no friction with a tiny mass and a 
large spring, we can ignore the mass.